© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a80 K11a82

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X22,10,1,9 X2,11,3,12 X18,13,19,14 X20,16,21,15 X8,17,9,18 X6,20,7,19 X16,22,17,21

Gauss Code:

{1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -5}

DT (Dowker-Thistlethwaite) Code:

4 10 12 14 22 2 18 20 8 6 16

Alexander Polynomial:

- t-4 + 6t-3 - 16t-2 + 26t-1 - 29 + 26t - 16t2 + 6t3 - t4

Conway Polynomial:

1 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a282, ...}

Determinant and Signature:

{127, 2}

Jones Polynomial:

- q-4 + 4q-3 - 8q-2 + 13q-1 - 17 + 20q - 20q2 + 18q3 - 13q4 + 8q5 - 4q6 + q7

Other knots (up to mirrors) with the same Jones Polynomial:

{K11a282, ...}

A2 (sl(3)) Invariant:

- q-12 + q-10 + q-8 - q-6 + 3q-4 - 3q-2 + 1 + q2 - 2q4 + 5q6 - 3q8 + 3q10 - q12 - 2q14 + 2q16 - 2q18 + q20

HOMFLY-PT Polynomial:

2a-4z2 + 3a-4z4 + a-4z6 - 6a-2z2 - 9a-2z4 - 5a-2z6 - a-2z8 + 1 + 6z2 + 7z4 + 2z6 - 2a2z2 - a2z4

Kauffman Polynomial:

a-8z4 - 2a-7z3 + 4a-7z5 + 2a-6z2 - 7a-6z4 + 8a-6z6 - a-5z + 5a-5z3 - 13a-5z5 + 11a-5z7 + a-4z2 + 2a-4z4 - 14a-4z6 + 11a-4z8 - 4a-3z + 19a-3z3 - 23a-3z5 - a-3z7 + 7a-3z9 - 9a-2z2 + 39a-2z4 - 50a-2z6 + 15a-2z8 + 2a-2z10 - 6a-1z + 18a-1z3 - 26a-1z7 + 12a-1z9 + 1 - 13z2 + 44z4 - 42z6 + 8z8 + 2z10 - 4az + 9az3 + 3az5 - 13az7 + 5az9 - 5a2z2 + 15a2z4 - 14a2z6 + 4a2z8 - a3z + 3a3z3 - 3a3z5 + a3z7

V2 and V3, the type 2 and 3 Vassiliev invariants:

{0, 0}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=2 is the signature of 1181. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                       1 j = 13                     3   j = 11                   5 1   j = 9                 8 3     j = 7               10 5       j = 5             10 8         j = 3           10 10           j = 1         8 11             j = -1       5 9               j = -3     3 8                 j = -5   1 5                   j = -7   3                     j = -9 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 81]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 81]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 7, 15, 8], > X[22, 10, 1, 9], X[2, 11, 3, 12], X[18, 13, 19, 14], X[20, 16, 21, 15], > X[8, 17, 9, 18], X[6, 20, 7, 19], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 81]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, > -8, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 81]]
Out[5]=	DTCode[4, 10, 12, 14, 22, 2, 18, 20, 8, 6, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 81]][t]
Out[6]=	-4 6 16 26 2 3 4 -29 - t + -- - -- + -- + 26 t - 16 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 81]][z]
Out[7]=	6 8 1 - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 81], Knot[11, Alternating, 282]}
In[9]:=	{KnotDet[Knot[11, Alternating, 81]], KnotSignature[Knot[11, Alternating, 81]]}
Out[9]=	{127, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 81]][q]
Out[10]=	-4 4 8 13 2 3 4 5 6 7 -17 - q + -- - -- + -- + 20 q - 20 q + 18 q - 13 q + 8 q - 4 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 81], Knot[11, Alternating, 282]}
In[12]:=	A2Invariant[Knot[11, Alternating, 81]][q]
Out[12]=	-12 -10 -8 -6 3 3 2 4 6 8 10 12 1 - q + q + q - q + -- - -- + q - 2 q + 5 q - 3 q + 3 q - q - 4 2 q q 14 16 18 20 > 2 q + 2 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 81]][a, z]
Out[13]=	2 2 4 4 6 2 2 z 6 z 2 2 4 3 z 9 z 2 4 6 z 1 + 6 z + ---- - ---- - 2 a z + 7 z + ---- - ---- - a z + 2 z + -- - 4 2 4 2 4 a a a a a 6 8 5 z z > ---- - -- 2 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 81]][a, z]
Out[14]=	2 2 2 3 z 4 z 6 z 3 2 2 z z 9 z 2 2 2 z 1 - -- - --- - --- - 4 a z - a z - 13 z + ---- + -- - ---- - 5 a z - ---- + 5 3 a 6 4 2 7 a a a a a a 3 3 3 4 4 4 5 z 19 z 18 z 3 3 3 4 z 7 z 2 z > ---- + ----- + ----- + 9 a z + 3 a z + 44 z + -- - ---- + ---- + 5 3 a 8 6 4 a a a a a 4 5 5 5 6 39 z 2 4 4 z 13 z 23 z 5 3 5 6 8 z > ----- + 15 a z + ---- - ----- - ----- + 3 a z - 3 a z - 42 z + ---- - 2 7 5 3 6 a a a a a 6 6 7 7 7 14 z 50 z 2 6 11 z z 26 z 7 3 7 8 > ----- - ----- - 14 a z + ----- - -- - ----- - 13 a z + a z + 8 z + 4 2 5 3 a a a a a 8 8 9 9 10 11 z 15 z 2 8 7 z 12 z 9 10 2 z > ----- + ----- + 4 a z + ---- + ----- + 5 a z + 2 z + ----- 4 2 3 a 2 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 81]], Vassiliev[3][Knot[11, Alternating, 81]]}
Out[15]=	{0, 0}
In[16]:=	Kh[Knot[11, Alternating, 81]][q, t]
Out[16]=	3 1 3 1 5 3 8 5 9 11 q + 10 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 8 q 3 5 5 2 7 2 7 3 9 3 > --- + 10 q t + 10 q t + 8 q t + 10 q t + 5 q t + 8 q t + t 9 4 11 4 11 5 13 5 15 6 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a81

K11a80 K11a82